where \(I_s\)  denotes the identity matrix of size any integer \(s\ge 0\) . The key property of a GLR-linearization is that it has the same finite eigenvalues with the same partial multiplicities as \(P(\lambda )\) .
Furthermore, \(L(\lambda )\)  is a strong linearization of \(P(\lambda )\)  in the GLR-sense if \(L(\lambda )\)  is a GLR-linearization of \(P(\lambda )\)  and \( \text{rev}_1L(\lambda )\)  is a GLR-linearization of \(\text{rev}_d P(\lambda )\) . Then, a GLR-strong linearization has the same finite and infinite eigenvalues with the same partial multiplicities as \(P(\lambda )\) . However, the minimal indices of a GLR (strong) linearization \(L(\lambda )\)  may be completely unrelated to those of \(P(\lambda )\)  [1]}, except for the fact that the number of left (resp. right) minimal indices of \(L(\lambda )\)  and \(P(\lambda )\)  are equal. Nevertheless, the GLR-strong linearizations that are used in practice have minimal indices that are simply related to those of the polynomial through addition of a constant shift (see [2]} and the references therein).
